10.11 Chapter Summary

Possible hypotheses with direction of extremes:

$Z$ -test on a single population mean:

• •

  The population $X$ must be normally distributed, or the sample size $n$ sufficiently large ($n\ge 30$).

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  The population standard deviation $\sigma$ must be known.

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  The test-statistic is the $z$-score of the sample mean $\overline{x},$ i.e., it is computed by

  $$z_0=\frac{\overline{x}-{\mu }_0}{\sigma /\sqrt{n}},$$

  where ${\mu }_0$ is the hypothesized value given in $H_0.$

• •

  The $p$-value is calculated as an area under the standard normal curve $N(0,1)$ determined by $z_0.$ The Excel command

  $$\mathrm{𝙽𝙾𝚁𝙼}.\mathrm{𝙳𝙸𝚂𝚃}({𝚉}_{\mathrm{𝟶}},\mathrm{𝟶},\mathrm{𝟷},\mathrm{𝚃𝚁𝚄𝙴})$$

  can be used to compute this area, but remember that this command computes the area to the left of $z_0.$

$T$ -test on a single population mean:

• •

  The population $X$ is normally distributed. (The $T$-Test is reliable if the population has a nearly normal distribution, or if the sample size is large and the distribution not too pathological.)

• •

  The test-statistic is the $t$-score of the sample mean $\overline{x},$ i.e., it is computed by

  $$t_0=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}},$$

  where ${\mu }_0$ is the hypothesized value given in $H_0.$

• •

  The $p$-value is calculated as an area under a $T$-distribution with $n-1$ degrees of freedom, determined by $t_0.$ The Excel command

  $$𝚃.\mathrm{𝙳𝙸𝚂𝚃}({𝚃}_{\mathrm{𝟶}},\mathrm{𝟶},\mathrm{𝟷},\mathrm{𝚃𝚁𝚄𝙴})$$

  can be used to compute this area, but remember that this command computes the area to the left of $t_0.$